The HLbL can be divided in three parts, which are the leading order parts of the HLbL piece [1]} shown in Fig. REF .
The sum of diagrams REF  (b) and REF  (c) are one order of magnitude suppressed as compared to the pseudoscalar exchange
shown in diagram REF  (a). We focus on the pseudoscalar pole contribution to the HLbL piece, \(a_\mu ^{P,HLbL}\) , which
in order to be fully described needs only the Transition Form Factor (TFF), \(\mathcal {F}_{P\gamma ^\star \gamma ^\star }(q^2,p^2)\) .
As it has been shown in ref [2]}, this TFF gives most of its contribution to \(a_\mu \)  at Euclidian squared photon momenta \(\lesssim 1\)  GeV\(^2\) .
Therefore, it will be saturated mainly with the lightest resonant part of the (TFF) and higher energies effects will be suppressed.
To compute the TFF we rely on \(\chi \) PT [3]}, [4]} extended to include the lightest multiplet of meson resonances [5]} (Resonance
Chiral Theory, R\(\chi \) T).
Instead of using the complete basis of VVP and VJP operators for resonances [6]}, we will rely on the simpler
basis given in [7]} since, for describing vertices involving only one pseudo-Goldstone both are
equivalent [8]}. We will use the Lagrangian in ref. [6]} to include effects due to pseudoscalar resonances.
The novelty in our approach is that we account for all the leading order terms that
break explicitly chiral symmetry, which enter as corrections in powers of the squared pseudo-Goldstone bosons, \(m_P^2\) .
Here we will only present those operators necessary for a consistent description of the NLO terms in \(m_P^2\) .
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